www.ijraset.com
IC Value: 13.98
Volume 3 Issue XI, November 2015
ISSN: 2321-9653
International Journal for Research in Applied Science & Engineering
Technology (IJRASET)
Finding an Optimal Solution of an Assignment
Problem by Improved Zero Suffix Method
S.Sudha1, Dr.D.Vanisri2
1,2
Assistant Professor, Department of Mathematics, Bharathiar University PG Extension Centre, Erode, Tamilnadu ,India
Abstract - In this paper improved Zero Suffix Method is applied for finding an optimal solution for assignment problem. This
method requires least iterations to reach optimality, compared to the existing methods available in the literature. Here numerical
examples are solved to check the validity of the proposed method.
Keywords - Assignment Problem, Optimal Solution, Linear Programming Problem, Cost minimization Assignment problem,
improved Zero suffix method.
I.
INTRODUCTION
The assignment problem is one of the earliest applications and is a special case of linear programming problem, which
deals with available sales-force to different regions: vehicles to routes; products to factories; contracts to bidders; machines to jobs;
development engineering to several construction sites and so on. Since all supplies, demands, and bounds on variables are integral,
the assignment problem relies on a nice property of transportation problems that the optimal solution will be entirely integral. As
you will see, the goal of the objective of the assignment problem (unlike transportation problems) may not have anything to do at all
with moving anything.Generally the management makes assignment on a one-to-one basis in such a manner that the group
maximizes the revenue from the sales; the vehicles are deployed to various routes in such a way that the assignment cost is
minimum and so on.Applications of assignment problems are varied in the real world. Certainly it can be useful for the classic task
of assigning employees to tasks or machines to production jobs, but its uses are more widespread. It could be used to assign fleets of
aircrafts to particular trips, or assigning school buses to routes, or networking computers. In rare cases, it can even be used to
determine marriage partners.A considerable number of methods have been so far presented for assignment problem in
which the Hungarian method is more convenient method among them. Different methods have been presented for transportation
problem and various articles have been published on the subject. Many of the authors [1] - [3] have done the transportation
problems in different methods. Pandian and Natarajan [4] proposed new method for finding an optimal solution directly for
transportation problem. An Optimal Solution for Transportation Problems solved by [5], [6] using zero suffix method and ASM
method respectively. The more for less method to distribution related problems was established by [7].
By a complete assignment for a cost matrix n × n, we mean an assignment plan containing exactly n assigned
independent zeros, one in each row and one in each column. The main concept of assignment problem is to find the optimum
allocation of a number of resources to an equal number of demand points. An assignment plan is optimal if optimizes the total cost
or effectiveness of doing all the jobs.
Hence we applied new method named improved zero suffix method, which is different from the preceding methods to apply in the
assignment problems.
II. IMPROVED ZERO SUFFIX METHOD
The working rule of finding the optimal solution is as follows:
Step 1: Construct the assignment problem.
Step2: Subtract each row entries of the assignment table from the row minimum element.
Step 3: Subtract each column entries of the assignment table from the column minimum element.
Step 4: In the reduced cost matrix there will be at least one zero in each row and column, and then find the suffix value of all the
zeros in the reduced cost matrix by following simplification, the suffix value is denoted by S.
S=
Step 5: Choose the maximum of S, if it has one maximum value then assign that task to the person and if it has more than one
maximum value then also assign the tasks to their respective persons (if the zeros don’t lie in the same column or row).
And if the zeros lie in the same row or column then assign the job to that person whose cost is minimum.
502
©IJRASET 2015: All Rights are Reserved
www.ijraset.com
IC Value: 13.98
Volume 3 Issue XI, November 2015
ISSN: 2321-9653
International Journal for Research in Applied Science & Engineering
Technology (IJRASET)
Now create a new assignment table by deleting that row & column which has been assigned.
Step 6: Repeat step 2 to step 3 until all the tasks has not been assigned to the persons.
A. Examples
(i)
Consider the following cost minimization assignment problem
J1
J2
J3
J4
B1
15
11
13
15
B2
17
12
12
13
B3
14
15
10
14
B4
16
18
11
17
Solution: On applying row minimum operation we get
J1
J2
J3
J4
B1
4
0
2
4
B2
5
0
0
1
B3
4
5
0
4
B4
5
7
0
6
Again on applying column minimum operation we get
J1
J2
J3
J4
B1
0
0
2
3
B2
1
0
0
0
B3
0
5
0
3
B4
1
7
0
5
Now find the suffix value of each element whose value is zero & write the suffix value within the bracket [ ]. The suffix value is
calculated by using the formula
S=
503
©IJRASET 2015: All Rights are Reserved
www.ijraset.com
IC Value: 13.98
Volume 3 Issue XI, November 2015
ISSN: 2321-9653
International Journal for Research in Applied Science & Engineering
Technology (IJRASET)
J1
J2
D1
J3
J4
D2
B1
0 [2.3]
0 [5.6]
B2
1
0 [3.25]
B3
0 [3.3]
B4
1
D3
D4
2
3
0 [0.6]
0 [4]
5
0 [2.5]
3
7
0
5
[5]
From all of the above suffix, 5.6 is the maximum so assign the job J2 to the person B1
Next delete the 1st row and 2nd column from the above table and apply the same process.
B2
B3
J1
J3
J4
1
0[0.25]
0[4.5]
0[2.5]
B4
0[0.75]
1
3
0[2]
5
From all of the above suffix, 4.5 is the maximum so assign the job J4 to the person B2
Next delete the 2nd row and 4th column from the above table and apply the same process.
J1
B3
J3
0[0]
0[0.5]
B4
1
0 [0.5]
From the above table, it is clear that the job J1 should be assigned to the person B3 and the job J3 should be assigned to the person
B4. Finally the assignments are as follows B1 → J2, B2 → J4, B3 → J1, B4 → J3
& the minimum assignment cost = Rs (11 + 13 + 14 + 11)
= Rs 49
(ii) Consider the following travelling salesman problem
504
©IJRASET 2015: All Rights are Reserved
www.ijraset.com
IC Value: 13.98
Volume 3 Issue XI, November 2015
ISSN: 2321-9653
International Journal for Research in Applied Science & Engineering
Technology (IJRASET)
O1
∞
30
0
24
O2
1
∞
10
0
O3
50
0
∞
28
O4
4
4
0
∞
Solution: On applying row minimum operation we get
D1
D2
D3
D4
O1
∞
30
0
24
O2
0
∞
10
0
O3
49
0
∞
28
O4
3
4
0
∞
Again on applying column minimum operation we get
D1
D2
D3
D4
O1
∞
46
16
40
O2
41
∞
50
40
O3
82
32
∞
60
O4
40
40
36
∞
Now find out the suffix value of each element whose value is zero & write the suffix value within the bracket [ ]. The suffix value
is calculated by using the formula
S=
D1
O1
D2
D3
D4
∞
30
0[32]
24
0[53]
∞
10
0[41]
49
0[111]
∞
28
3
4
36[8.5]
∞
O2
O3
O4
505
©IJRASET 2015: All Rights are Reserved
www.ijraset.com
IC Value: 13.98
Volume 3 Issue XI, November 2015
ISSN: 2321-9653
International Journal for Research in Applied Science & Engineering
Technology (IJRASET)
From all of the above suffix, 111 is maximum, so assign the origin O3 to the Destination D2. Next delete the 3rd row and 2nd column
from above table and repeat the same process
D1
D3
D4
O1
∞
0[17]
24
O2
0[13.5]
10
0[12]
O4
3
0[6.5]
∞
D1
D3
D4
O1
∞
0
24
O2
0
10
0
O4
3
0
∞
Now find out the suffix value of each element whose value is zero
From all of the above suffix, 17 is maximum, so assign the origin O1 to the Destination D3
Next delete the 1st row and 3rd column from above table and repeat the same process
D1
D4
O2
∞
0[28]
O4
0[28]
28
From the above table it is clear that O2 → D4, O4 → D1
The optimum assignment is O1 → D3, O2 → D4, O3 → D2, O4 → D1
& the minimum cost = Rs (16 + 40 + 32 + 40 )
=Rs 128
III.CONCLUSION
The proposed algorithm carries systematic procedure, and very easy to understand. From this paper, it can be concluded
that Improved zero suffix Method provides an optimal solution in fewer iterations, for the assignment problems. As this method
consumes less time and is very easy to understand and apply, so it will be very helpful for decision makers.
REFERENCES
[1] A. Edward Samuel and M. Venkatachalapathy, “Modified Vogel’s Approximation method for Fuzzy Transportation Problem”, Applied Mathematical
Science, Vol.5, No.28, pp.1367-1372, 2011.
[2] NagarajBalakrishnan, “Modified Vogel’s Approximation Method for Unbalanced Transportation Problem”, Applied Mathematics Letters 3, pp.9-11, 1990.
[3] A. Nagooragani, “Fuzzy Transporation problem, Proceedings of national Seminar on Recent Advancement in Mathematics, 2009.
[4] P. Pandian, and G. Natarajan, “A new method for finding an optimal solution for Transportation problem,” International Journal of Mathematical
Sciences and Engineering Applications, Vol. 4, pp. 59-65, 2010.
506
©IJRASET 2015: All Rights are Reserved
www.ijraset.com
IC Value: 13.98
Volume 3 Issue XI, November 2015
ISSN: 2321-9653
International Journal for Research in Applied Science & Engineering
Technology (IJRASET)
[5] V. J. Sudhakar, N. Arunsankar, and T. Karpagam, “A New approach for finding an optimal Solution for transportation problem”, Applied European
Journal of Scientific Research,Vol.68, pp. 254-257, 2012.
[6] Abdul Quddoos, ShakeelJavaid, and M. M. Khalid, “A New method for finding an Optimal Solution for Transportation Problem”, International Journal on
Computer Science and Engineering, Vol.4, No.7, July 2012.
[7] VeenaAdlakha, and Krzysztof Kowalski, “A heuristic method for more-for less in distribution related problems”, Internal Journal of Mathematical
Education in Science and Technology, Vol. 32, pp. 61-71, 2001.
[8] H. A. Taha, Operations Research- Introduction, Prentice Hall of India, New Delhi, 2004.
[9] J. K. Sharma, Operations Research- Theory and applications, Macmillan India LTD, New Delhi, 2005.
[10] KantiSwarup, P. K. Gupta and Man Mohan, Operations Research, Sultan Chand & Sons, 12th Edition, 2004.
[11] P. K. Gupta, D. S. Hira, Operations Research, S. Chand & Company Limited, 14th Edition, 1999.
507
©IJRASET 2015: All Rights are Reserved